Can a number be both rational and irrational? And what about numbers like $\pi$ or $e$? Are they always irrational?

No, a number cannot be both rational and irrational. These two categories are mutually exclusive and exhaustive for real numbers. A rational number can be expressed as a fraction $p/q$ where $p$ and $q$ are integers and $q \ne 0$, and its decimal expansion is either terminating or non-terminating repeating. An irrational number, conversely, cannot be expressed in this fractional form, and its decimal expansion is non-terminating and non-repeating. Numbers like $\pi$ (pi) and $e$ (Euler's number) are indeed always irrational. Their decimal expansions continue infinitely without any repeating pattern. This is a fundamental property of these specific mathematical constants, and their irrationality has been rigorously proven.